Just finding the water being down, indicating a slow leak from the tank, is probably your only real problem. It could be more than just the flapper, but it's 95% likely that it's the flapper. So...if you're in Manhattan just go to Rainbow on 1st Ave on the west side between 75th and 76th (it's a fun hardware/everything store), which has an excellent selection of Korky flappers. They should have the Korky 3060BP large 3" adjustable flapper, which is a quick and easy replacement (or you can get it from that big online store that also sells the Kindle). The 2021BP "Fits Toto" GMax flapper will also work, but the latter will stay open a little longer than optimal, eliminating some of the water savings, making it more like a 1.8gpf toilet (more water in flush, longer refill and some refill water wasted down the drain). Korky doesn't list the setting for the Drake II in its current instructions on the adjustable 3" flapper, but you can call them (they're really nice) and they'll tell you, or you can use the method of marking the water level when the old flapper closes and adjusting the white band so that it mimics that. Or you can wing it because at the end of the day it doesn't make *that* much difference. (There are other places in Manhattan than Rainbow to find Korky products, but that one is in my neighborhood and has a zillion Korky products. If you are outside of the City, I would just get the 3060PK at the nearest Lowe's. Lowe's uses variations on the Korky model number on its web site, so it's 3060PK at Lowe's and 3060BP everywhere else.)
Or call Toto and tell them to send you a new flapper because it quit way too soon. That will take about a week to arrive.
As to the water dripping from the fill valve, I wouldn't sweat it. On many anti-siphon fill valves, there is always a little water running from the valve head. It goes into the tank anyway, so it's not an issue. If you look at the instructions for the Korky replacement valve, they make an explicit point that water is expected to flow from multiple places on the valve, including about the same place that you're pointing to on the Voreto. (Korky is the original valve on a lot of Toto toilets, but not yours; you have the Voreto fill valve, which is interchangeable with the Korky 528MP or the Toto TSU99A.X Universal Replacement Fill Valve, which is made by Korky in Wisconsin.)
I'm guessing that the chattering that you are hearing is the fill valve opening a little to replace water that has escaped the tank by dripping past the flapper, and then shutting off. When you eliminate the leak from the flush valve and flapper, you won't need to worry about the fill valve.
Finally, I would rinse off that screen; it appears to have a little schmutz on the top part of the photo, if I'm viewing correctly.
Let us know how it turns out.